and we must relate \(\chi (X : \rho ^{t}_{\operatorname{{\mathcal {E}}}_X})\)  to \(\chi (X : \rho ^{t-1}_{\operatorname{{\mathcal {E}}}_X})\) .
To achieve this goal, we use the fact that the Holevo-\(\chi \)  is closely related to the quantum relative entropy \(D(\rho || \sigma ) = \operatorname{tr}\left( \rho (\log \rho - \log \sigma ) \right)\)  [1]}, [2]}, [3]}. Indeed,
\(\chi (X : \rho ^t_{\operatorname{{\mathcal {E}}}_X})= \operatornamewithlimits{{\mathbb {E}}}_X\left[ \operatorname{tr}\left( \rho ^t_{\operatorname{{\mathcal {E}}}_X} \log \rho ^t_{cE_X} - \rho ^t_{\operatorname{{\mathcal {E}}}_X} \log \left(\operatornamewithlimits{{\mathbb {E}}}_{X^{\prime }}\rho ^t_{cE_{X^{\prime }}} \right) \right) \right]= \operatornamewithlimits{{\mathbb {E}}}_{X} D\left(\rho ^t_{\operatorname{{\mathcal {E}}}_X} || \operatornamewithlimits{{\mathbb {E}}}_{X^{\prime }}\rho ^t_{cE_{X^{\prime }}}\right),\) 
